code: def _get_pdA2(
      results, tape, pdA2_fn, non_involutory_indices, var_indices
  ):
      """The main auxiliary function to get the partial derivative of <A^2>."""
      pdA2 = 0
  
      if non_involutory_indices:
          # compute the second derivative of non-involutory observables
          pdA2 = pdA2_fn(results)
  
          # For involutory observables (A^2 = I) we have d<A^2>/dp = 0.
          if involutory := set(var_indices) - set(
              non_involutory_indices
          ):
              if tape.shots.has_partitioned_shots:
                  pdA2 = tuple(
                      _put_zeros_in_pdA2_involutory(
                          tape, pdA2_shot_comp, involutory
                      )
                      for pdA2_shot_comp in pdA2
                  )
              else:
                  pdA2 = _put_zeros_in_pdA2_involutory(
                      tape, pdA2, involutory
                  )
      return pdA2
  
imports: from typing import Sequence, Callable
  from functools import partial
  import numpy
  import pennylane
  from pennylane.measurements import VarianceMP
  from pennylane import transform
  from pennylane.transforms.tape_expand import expand_invalid_trainable
  from pennylane.gradients.gradient_transform import _contract_qjac_with_cjac
  from .finite_difference import finite_diff
  from .general_shift_rules import _iterate_shift_rule, frequencies_to_period, generate_shifted_tapes, process_shifts
  from .gradient_transform import _all_zero_grad, assert_no_state_returns, assert_no_tape_batching, assert_multimeasure_not_broadcasted, choose_trainable_params, find_and_validate_gradient_methods, _no_trainable_grad, reorder_grads
label_desc: Use image node_id 13 for calling a global function with example usage: _get_pdA2(results, tape, pdA2_fn, non_involutory_indices, var_indices) and returns: pdA2